Similar Rectangles

Date: 05/03/97 at 19:05:44
From: Emily
Subject: Similarity

This problem is driving me crazy because everything I try produces an 
incorrect answer. 

Presto Printing makes cards which, when opened, have an outside 
boundary similar to that when they are folded. Find the width w of the 
card if the length when open is 8 and when folded is 4. 

I am also not entirely clear on what the question means! I know that 
the correct answer is the square root of 32, but I have no idea why.

Date: 05/04/97 at 03:26:59
From: Doctor Mike
Subject: Re: Similarity

Hello Emily,    
  
Let's start with what similarity means. 

Saying two rectangles are similar means that the ratio of the longer 
dimension to the shorter dimension is the same for both. For example, 
if the ratio is 2:1 or 2-to-1, then possible similar rectangles could 
have the dimensions 3 miles by 1.5 miles, 3 meters by 6 meters, or 
1000 meters by 500 meters. If you form a fraction with the longer 
dimension over the shorter one, the value is always the same; that is, 
2/1 and 3/1.5 and 6/3 and 1000/500 all equal two. 
  
Now I will try to draw the greeting card both closed and open: 
        __________              ______________________
        |         |             |          |          |
        |         |             |                     |
        |         |             |          |          |
      w |         |             |                     | w
        |         |             |          |          |
        |         |             |                     |
        |_________|             |__________|__________|
             4                             8   

The rectangular shape on the left for the closed card has a larger
vertical dimension, and the rectangle shape on the right for the 
opened card has a larger horizontal dimension. (Note, that the dashed 
vertical line in the center of the diagram on the right is the fold-
line.)  So, the ratio of longer-to-shorter is w:4 on the left and 8:w 
on the right.  If we express these ratios as fractions, w/4 and 8/w, 
then we can solve by algebra for the value of w making the 2 fractions 
equal.
         
           w       8    
          ---  =  ---  
           4       w     
  
Solve this equation for w to get the answer.  I think you can
take it from here.  I hope this helps. 
      
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/